Prerequisites: Score of 550 on the mathematics portion of the SAT completed within the last year or the appropriate gade on the General Studies Mathematics Placement Examination.

For students who wish to study calculus but do not know analytic geometry. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits.

Prerequisites: see Courses for First-Year Students. Functions, limits, derivatives, introduction to integrals.

The Help Room on the 3rd floor of Milbank Hall (Barnard College) is open during the day, Monday through Friday, to students seeking individual help from the instructors and teaching assistants. (SC)

Prerequisites: MATH V1101 or the equivalent.

Methods of integration, applications of the integral, Taylor's theorem, infinite series. (SC)

Prerequisites: MATH V1101 with a grade of B or better or Math V1102, or the equivalent.

Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC)

Prerequisites: MATH V1102, V1201, or the equivalent.

Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)

Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form. Intended for students who are considering majoring in mathematics but wish additional training.

Prerequisites: Some calculus or permission of the instructor.

Intended as an enrichment to the mathematics curriculum of the first two years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.

Prerequisites: V1201, or the equivalent.

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)

Prerequisites: Math V1102-Math V1201 or the equivalent and MATH V2010.

Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (SC)

Prerequisites: Calculus I, II, III and Linear Algebra.

A concrete introduction to abstract algebra. Topics in abstract algebra used in cryptography and coding theory.

Prerequisites: MATH V1201 or the equivalent.
Corequisites: MATH V2010.

Equations of order one; systems of linear equations. Second-order equations. Series solutions at regular and singular points. Boundary value problems. Selected applications.

Prerequisites: MATH V1202 or the equivalent.

Local and global differential geometry of submanifolds of Euclidiean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem.

Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the permission of the director of undergraduate studies.

The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow.

Prerequisites: The written permission of the faculty member who agrees to act as a supervisor, and the permission of the director of the undergraduate studies.

For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member. .

The second term of this course may not be taken without the first. Prerequisite: Math V1102-Math V1202 and MATH V2010, or the equivalent. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.

Prerequisites: MATH W4041-W4042 or the equivalent.

Algebraic number fields, unique factorization of ideals in the ring of algebraic integers in the field into prime ideals. Dirichlet unit theorem, finiteness of the class number, ramification. If time permits, p-adic numbers and Dedekind zeta function.

Prerequisites: MATH V1202, MATH V2010, and rudiments of group theory (e.g., MATH W4041). MATH V1208 or W4061 is recommended, but not required.

Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces.

Prerequisites: The second term of this course may not be taken without the first. Prerequisites: MATH V1202 or the equivalent and V2010.

Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

Prerequisites: MATH V1207 and Math V1208 or MATH W4061.

A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta function, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory.

Prerequisites: MATH V1202, V3027, STAT W4150, SEIO W4150, or their equivalents.

The mathematics of finance, principally the problem of pricing of derivative securities, developed using only calculus and basic probability. Topics include mathematical models for financial instruments, Brownian motion, normal and lognormal distributions, the BlackûScholes formula, and binomial models.

Topics include holomorphic functions; analytic continuation; Riemann surfaces; theta functions and modular forms.

Commutative rings; modules; localization; primary decoposition; integral extensions; Noetherian and Artinian rings; Nullstellensatz; Dedekind domains; dimension theory; regular local rings.

Topics include homology and homotopy theory; covering spaces; homology with local coefficients; cohomology; Chech cohomology.

Topics include basic notions of groups with algebraic and geometric examples; symmetry; Lie algebras and groups; representations of finite and compact Lie groups; finite groups and counting principles; maximal tori of a compact Lie group.

Prerequisites: MATH V1202 or the equivalent and MATH V2010.

This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.

Prerequisites: MATH V1201 or the equivalent.

Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

Required for all applied mathematics major in the junior year. Prerequisites or corequisites: Math V3027, V3028, and V2010, or their equivalents. Introductory seminars on problems and techniques in applied mathematics. Typical topics are nonlinear dynamics, scientific computation, economics, operation research, etc.

Prerequisites or corequisites: MATH V3007, V3028, and V2010, or their equivalents. Required for all applied mathematics majors in the senior year. It consists of the same weekly lecture as APMA E4901 plus two hours of tutorials per week. Examples of problem areas are nonlinear dynamics, asymptotics, approximation theory, numerical methods, etc.

Prerequisites: Score of 550 on the mathematics portion of the SAT completed within the last year or the appropriate gade on the General Studies Mathematics Placement Examination.

For students who wish to study calculus but do not know analytic geometry. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits.

Prerequisites: see Courses for First-Year Students. Functions, limits, derivatives, introduction to integrals.

The Help Room on the 3rd floor of Milbank Hall (Barnard College) is open during the day, Monday through Friday, to students seeking individual help from the instructors and teaching assistants. (SC)

Prerequisites: MATH V1101 or the equivalent.

Methods of integration, applications of the integral, Taylor's theorem, infinite series. (SC)

Prerequisites: MATH V1101 with a grade of B or better or Math V1102, or the equivalent.

Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC)

Prerequisites: MATH V1102, V1201, or the equivalent.

Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)

Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form. Intended for students who are considering majoring in mathematics but wish additional training.

Prerequisites: V1201, or the equivalent.

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)

Prerequisites: Math V1201

A more extensive treatment of the material in Math V2010, with increased emphasis on proof. Not to be taken in addition to Math V2010 or Math V1207-V1208.

Prerequisites: Math V1102-Math V1201 or the equivalent and MATH V2010.

Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (SC)

Prerequisites: MATH V1202. An elementary course in functions of a complex variable.

Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations. Cauchy integral theorem. Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.(SC)

Prerequisites: one year of calculus.

Congruences. Primitive roots. Quadratic residues. Contemporary applications.

Prerequisites: MATH V3027 and MATH V2010 or the equivalent

. Introduction to partial differential equations. First-order equations. Linear second-order equations; separation of variables, solution by series expansions. Boundary value problems.

Prerequisites: MATH V1102, V1201(or V1101, V1102, V1201), V2010. Recommended: MATH V3027(or MATH E1210) and SIEO W3600.

Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates.

Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the permission of the director of undergraduate studies.

The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow.

Prerequisites: The written permission of the faculty member who agrees to act as a supervisor, and the permission of the director of the undergraduate studies.

For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member. .

Prerequisites: three terms of calculus and linear algebra or four terms of calculus.

Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines.

The second term of this course may not be taken without the first. Prerequisite: Math V1102-Math V1202 and MATH V2010, or the equivalent. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.

Prerequisites: Math V2010 or equivalent, Math W4041 and Math W4051.

The study of algebraic and geometric properties of knots in R^3, including but not limited to knot projections and Reidemeister's theorm, Seifert surfaces, braids, tangles, knot polynomials, fundamental group of knot complements. Depending on time and student interest, we will discuss more advanced topics like knot concordance, relationship to 3-manifold topology, other algebraic knot invariants.

Prerequisites: MATH V21010, MATH W4041, MATH W4051

The study of topological spaces from algebraic properties, including the essentials of homology and the fundamental group. The Brouwer fixed point theorem. The homology of surfaces. Covering spaces.

Prerequisites: The second term of this course may not be taken without the first. Prerequisites: MATH V1202 or the equivalent and V2010.

Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

Prerequisites: MATH W4051 or W4061 and V2010.

The implicit function theorem. Concept of a differentiable manifold. Tangent space and tangent bundle, vector fields, differentiable forms. Stoke's theorem, tensors. Introduction to Lie groups.

Prerequisites: MATH W4061 or MATH V3007.

A rigorous introduction to the concepts and methods of mathematical probability starting with basic notions and making use of combinatorial and analytic techniques. Generating functions. Convergence in probability and in distribution. Discrete probability spaces, recurrence and transience of random walks. Infinite models, proof of the law of large numbers and the central limit theorem. Markov chains.

Continuation of Mathematics G4175 (see Fall listing).

Affine and projective varieties; schemes; morphisms; sheaves; divisors; cohomology theory; curves; Riemann-Roch theorem.

Continuation of Mathematics G4307 (see Fall listing).

Continuation of Mathematics G4343 (see Fall listing).

Prerequisites: MATH V1202 or the equivalent and MATH V2010.

This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.

Prerequisites: MATH V1201 or the equivalent.

Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

Required for all applied mathematics major in the junior year. Prerequisites or corequisites: Math V3027, V3028, and V2010, or their equivalents. Introductory seminars on problems and techniques in applied mathematics. Typical topics are nonlinear dynamics, scientific computation, economics, operation research, etc.

Prerequisites or corequisites: MATH V3007, V3028, and V2010, or their equivalents. Required for all applied mathematics majors in the senior year. It consists of the same weekly lecture as APMA E4901 plus two hours of tutorials per week. Examples of problem areas are nonlinear dynamics, asymptotics, approximation theory, numerical methods, etc.